# Prove To Gavin Jones That The Results He Obtained Were Not Accidents

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If everybody uses the mean-variance approach to investing, and if everybody has the same estimates of the asset's expected returns, variances, and covariances, then everybody must invest in the same fund F of risky assets and in the risk-free asset Because F is the same for everybody, it follows that, in equilibrium, F must correspond to the market portfolio M—the portfolio in which each asset is weighted by its proportion of total market capitalization. This observation is the basis for the capital asset pricing model (CAPM).

If the market portfolio M is the efficient portfolio of risky assets, it follows that the efficient frontier in the r~a diagram is a straight line that emanates from the risk-free point and passes through the point representing M. This line is the capital market line. Its slope is called the market price of risk. Any efficient portfolio must lie on this line .

The CAPM is derived directly from the condition that the market portfolio is a point on the edge of the feasible region that is tangent to the capital market line; in other words, the CAPM expresses the tangency conditions in mathematical form The CAPM result states that the expected rate of return of any asset i satisfies

where fij ~ cov(/-;,) M)/oh is the beta of the asset

The CAPM can be represented graphically as a security market line: the expected rate of return of an asset is a straight-line function of its beta (or, alternatively, of its covariance with the market); greater beta implies greater expected return Indeed, from the CAPM view ir follows that the risk of an asset is fully characterized by its beta It follows, for example, that an asset that is uncorrected with the market (fi = 0) wiil have an expected rate of return equal to the risk-free rate

The beta of the market portfolio is by definition equal to 1. The betas of other stocks take other values, but the betas of most U S stocks range between .5 and 2 5

The beta of a portfolio of stocks is equal to the weighted average of the betas of the individual assets that make up the portfolio

One application of CAPM is to the evaluation of mutual fund performance, The Jensen index measures the historical deviation of a fund from the security market line (This measure has dubious value for funds of publicly traded stocks, however.) The Sharpe index measures the slope of the line joining the fund and the risk-free asset on the 7~<x diagram, so that this slope can be compared with the market price of risk

The CAPM can be converted to an explicit formula for the price of an asset .In the simplest version, this formula states that price is obtained by discounting the expected payoff, but the interest rate used for discounting must be ij + P(r,\f — iy), where is the beta of the asset An alternative form expresses the price as a discounting of the certainty equivalent of the payoff, and in this formula the discounting is based on the risk-free rate /y.

It is important to recognize that the pricing formula of CAPM is linear, meaning that the price of a sum of assets is the sum of their prices, and the pr ice of a multiple of an asset is that same multiple of the basic price The certainty equivalent formulation of the CAPM clearly exhibits this linear property.

The CAPM can be used to evaluate single-period projects within firms. Managers of firms should maximize the net present value of the firm, as calculated using the pricing form of the CAPM formula. This policy will generate the greatest wealth for existing owners and provide the maximum expansion of the efficient frontier for all mean-variance investors.

### EXERCISES

(Capital market line) Assume that the expected rate of return on the market portfolio is 23% and the rate of return on T-bilis (the risk-free rate) is 7%. The standard deviation of the market is ,32% Assume that the market portfolio is efficient

(a) What is the equation of the capital market line?

(b) (i) If an expected return of 39% is desired, what is the standard deviation of this position? (ii) If you have \$ 1,000 to invest, how should you allocate it to achieve the above position?

(c) If you invest \$300 in the risk-free asset and \$700 in the market portfolio, how much money should you expect to have at the end of the year?

2. (A small world) Consider a world in which there are only two risky assets, A and B, and a risk-free asset f The two risky assets are in equal supply in the market; that is, M = B) The following information is known: i /. = .10, a\ = 04, aA/, — 01, a\ = 02. and rA, = 18

(a) Find a general expression (without substituting values) for rr^, and pa

(b) According to the CAPM, what are the numerical values of r,\ and 7V?

3. (Bounds on returns) Consider a universe of just three securities They have expected rates of return of 10%, 20%, and 10%, respectively Two portfolios are known to lie on the minimum-variance set. They are defined by the portfolio weights

It is also known that the market portfolio is efficient,

(a) Given this information, what are the minimum and maximum possible values for the expected rate of return on the market portfolio? {b) Now suppose you are told that w represents the minimum-variance portfolio, Does this change your answers to part (£))?

4. (Quick CAPM derivation) Derive the CAPM formula for rk —r/ by using Equation (6 9) in Chapter 6 [Hint: Note that n

Apply (6 9) both to asset k and to the market itself

5, (Uncorrelated assets) Suppose there are n mutually unconelated assets The return on asset i has variance erf The expected rates of return are unspecified at this point, The total amount of asset i in the market is Xi ■ We let T = and then set Xi — Xj/T7 for / — 1,2, . ,/r Hence the market portfolio in normalized form is x = (,V|,.v2, . ,.v„) Assume there is a risk-free asset with rate of return /y Find an expression for ^ in terms of the a'/'s and a,-'s

6. (Simpleland) In Simpleland there are only two risky stocks, A and B, whose details are listed in Table 7.4

TABLE 7A

Details of Stocks A and B

Number of shares Price Expected Standard deviation outstanding per share rate of relurn of return

Furthermore, the correlation coefficient between the returns of stocks A and B is pAn = ^ There is also a risk-free asset, and Simpleland satisfies the CAPM exactly

(a) What is the expected rale of return of the market portfolio?

(b) What is the standard deviation of the market portfolio?

(d) What is the risk-free rate in Simpleland?

7. (Zero-beta assets) Let w0 be the portfolio (weights) of risky assets corresponding the minimum-variance point in the feasible region Let W| be any other portfolio on the efficient frontier Define / n and / | to be the corresponding returns

(a) There is a formula of the form ct()i = Acr^ Find A. [Hint Consider the portfolios (1 - a)vv() -f aw,, and consider small variations of the variance of such portfolios near a s 0]

(b) Corresponding to the portfolio Wj there is a portfolio w: on the minimum-variance set that has zero beta with respect to wf, that is, - = 0 This portfolio can be expressed as w. = (1 ~ a)w0 + aW| Find the proper value of of.

(c) Show the relation of the three portfolios on a diagram that includes the feasible region

(d) If there is no risk-free asset, it can be shown that other assets can be priced according to the formula where the subscript M denotes the market portfolio and r. is the expected rate of return on the portfolio that has zero beta with the market portfolio. Suppose that the expected returns on the market and the zero-beta portfolio are 15% and 9%, respectively Suppose that a stock i has a correlation coefficient with the market of .5 Assume also that the standard deviation of the returns of the market and stock i are 15% and 5%, respectively Find the expected return of stock i

8. (Wizards o) Electron Wizards, Inc (EW1) has a new idea for producing TV sets, and it is planning to enter the development stage Once the product is developed (which will be at the end of 1 year), the company expects to sell its new process for a price p, with expected value ~p = \$24M However, this sale price will depend on the market for T V sets at the time By examining the stock histories of various TV companies, it is determined that the final sales price p is correlated with the market return as E[(p ~~p){tst —O/)] = \$20Ma^ To develop the process, EWI must invest in a research and development project The cost c of this project will be known shortly after the project is begun (when a technical uncertainty will be resolved) The current estimate is that the cost will be either c = \$20M or c = S16M, and each of these is equally likely (This uncertainty is uncorrected with the final price and is also uncorrelated with the market) Assume that the risk-free rate is jy = 9% and the expected return on the market is 7,\t — 33%

(a) What is the expected rate of return of this project?

(b) What is the beta of this project? [Hint- In this case, note that

(t) Is this an acceptable project based on a CAPM criterion? In particular, what is the excess rate of return (4- or —) above the return predicted by the CAPM?

9„ (Gavin's problem) Prove to Gavin Jones that the results he obtained in Examples 7 5 and 7.7 were not accidents Specifically, for a fund with return ary + (1 — ct)>M, show that both CAPM pricing formulas give the price of \$100 worth of fund assets as \$100

The CAPM theory was developed independently in references 11—4] There are now numerous extensions and textbook accounts of that theory Consult any of the basic finance textbooks listed as references for Chapter 2 The application of this theory to mutual fund performance evaluation was presented in [5, 6] An alternative measure, not discussed in this chapter, is due n - rz ~ j8fM (*",»/

### REFERENCES

to Treynor . For summaries of the application of CAPM to corporate analysis, see f8, 9] The idea of using a zero-beta asset, as in Exercise 7, is due to Biack 

1 Sharpe, W. F. (1964), "Capital Asset Prices: A Theory of Market Equilibrium under Con ditions of Risk," Journal of Finance, 19, 425-442

2 L imner, J (1965), "The Valuation of Risk Assets and the Selection of Risky Investment in

Stock Portfolios and Capital Budgets," Review of Economics and Statistics, 47, 1.3—37

3 Mossin, J (1966), "Equilibrium in a Capital Asset Market," Economctrica, 34, no. 4,

768-783

4 Treynor, J L (1961), "Towards a Theory of Market Value of Risky Assets," unpublished manuscript

5. Sharpe, W F (1966), "Mutual Fund Performance." Journal of Business, 39, January,

119-138

6. Jensen, M C (1969), "Risk, the Pricing of Capital Assets, and the Evaluation of Investment

Portfolios," Journal of Business, 42, April, 167-247.

7. Treynor, J L. (1965), "How to Rate Management Investment Funds," Harvard Business

Review, 43, January-February, 63-75 8 Rubinstein, M. E (1973), "A Mean-Variance Synthesis of Corporate Financial Theory,"

Journal of Finance, 28, 167-182 9. Fama, E F (1977), "Risk-Adjusted Discount Rates and Capital Budgeting under Uncertainty," Journal of Financial Economics, 5, 3-24 10 Black, F (1972), "Capital Market Equilibrium with Restricted Borrowing," Journal of Business, 45, 445-454 